Despite the fact that fractional calculus has a lengthy history in mathematics, it has only recently seen a considerable number of real-world applications. Fractional derivatives are important due to their attractive characteristics (e.g. memory effect), including their wide dynamical range [1]}, [2]}, [3]}.
Many definitions of fractional derivatives and integrals exist, including Riemann-Liouville, Caputo, Grunwald-Letnikov [4]}. In 2015, Caputo and Fabrizio (CF) have suggested a new idea of fractional derivatives based on the exponential decay [5]}. Following that, Atangana and Baleanu (AB) proposed a novel concept of fractional derivatives with non-singular and non-local kernel based on the Mittag-Leffler function [6]}.
Due to the difficulty or impossibility of obtaining to obtain exact solution of fractional differential equations, numerical techniques are required to provide approximate solutions. Over the past few decades, numerical methods to solve initial value problems have attracted great interest from the research community,
which support various disciplines, including medicine, chemistry, biology, physics, business, and the arts, etc. Where computational algorithms are implemented. Moreover, numerical analysis techniques are the perfect tools to evaluate the behavior of real-life complicated models. Therefore, the implementation of accurate numerical methods is suibtle in the problems representing real world phenomena [6]}, [8]}, [9]}, [10]}. Over 2000 years ago, the simplest method in the literature was introduced with the concept of interpolation. In contrast, several other techniques were introduced for nonlinear systems, such as Newton’s method, Euler’s method, Gaussian elimination, and Lagrange interpolation [11]}, [12]}, [13]}, [14]}. Over the past few years, several numerical tools have been developed for solving fractional ordinary differential equations with general nonlinearity [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}. Such as the fractional Adams-Bashforth method, which is the most popular scheme in which the basis of Lagrange interpolation is used [22]}, [23]} and the fractional Adams-Bashforth/Moulton method developed with Newton linear/quadratic interpolations [24]}, [25]}.
Baleanu et al. [18]} used the definition of the Atangana-Baleanu derivative in the Caputo sense to produce a predictor-corrector approach for fractional differential equation as well as to prove the existence and uniqueness of such problem. They have converted the ABC-fractional ordinary differential equation into a Volterra Integral equation, then they have used the rectangle rule for the predictor and the trapezoid rule for the corrector.
On the other hand, based on the techniques and approximations appeared in [19]}, [20]} for finding approximate solutions of fractional differential equations with Caputo and Caputo-Fabrizio fractional derivatives, respectively. We can to propose a new and efficient predictor-corrector method for solving ABC-fractional initial value problems by using quadratic lagrange interpolation, memory term, and second-order Taylor’s expansion.
